3. Sequences of Numbers

3.4. Lim Sup and Lim Inf

When dealing with sequences there are two choices:

the sequence converges

the sequence diverges

While we know how to deal with convergent sequences, we don't know much about
divergent sequences. One possibility is to try and extract a convergent
subsequence, as described in the last section. In particular,
Bolzano-Weierstrass' theorem can be useful in case the original sequence was
bounded. However, we often would like to discuss the limit of a sequence
without having to spend much time on investigating convergence, or thinking
about which subsequence to extract. Therefore, we need to broaden our concept
of limits to allow for the possibility of divergent sequences.

and let c = lim (Aj). Then c is called
the limit inferior of the sequence
.

Let be a sequence of real
numbers. Define

Bj = sup{aj , aj + 1 , aj + 2 , ...}

and let c = lim (Bj). Then c is called
the limit superior of the sequence
.

In short, we have:

lim inf(aj) = lim(Aj) , where
Aj = inf{aj , aj + 1 , aj + 2 , ...}

lim sup(aj) = lim(Bj) , where
Bj = sup{aj , aj + 1 , aj + 2 , ...}

When trying to find lim sup and lim inf for a given sequence, it is best to
find the first few Aj's or Bj's,
respectively, and then to determine the limit of those. If you try to guess the
answer quickly, you might get confused between an ordinary supremum and the
lim sup, or the regular infimum and the lim inf.

The first equation is a conjecture, not yet proven, called the twin prime
conjecture. In fact, it is not even known if the lim inf is finite.
On the other hand, the second equation involving lim sup is known to be
infinite because of arbitrary spaces between two primes.